Don't do it.
Find a home boarder.
Cancel.
Those are your options.
You think your dog doesn't bark, but chances are in an unfamiliar environment of a caravan, all alone, strange noises going on outside, your dog WILL bark.
Typically, caravan/holiday cottages have a strict rule about not leaving dogs unattended in the property - precisely because even dogs that are usually fine at home and don't normally bark, are much more likely to do so in an unfamiliar environment (and of course a much higher risk of damage to the property).
So even if you did have a dog friendly van, you still would not be able to go out in the evening and leave your dog in the van.
Other people have explained the allergy thing - supposedly non-shedding dogs have a lower chance of triggering allergies, but they are not 100% - they do shed hair, skin and saliva particles.
All dogs have to shed hair - hair grows, hair dies, hair comes out of the follicles. But some breeds shed very little, and their coat type means this doesn't fall out much until you brush it out (poodles for example) and some shed copiously and it falls out all over the place and some are in between.
And everyone leaves other particles like skin, saliva, fecal matter, wherever they go (dog, human, orangutan, we all do it.)
You're not doing yourselves, your kids, your dog or anyone else using that van after you, any favours here.